Download Mathematical Analysis, Applications, and Connections

Mathematical Analysis, Applications, and Connections
Course Outline Alignment
(Text: A Survey of Mathematics with Applications: Angel, Abbot, Rude)
QUARTER 1 – Algebra Review
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Chapter 5: Number Theory and the Real Number System (N.RN)
Chapter 6: Algebra, Graphs, and Functions (A.APR, A.CED, F.IF, F.BF, F.LE)
Chapter 7: Systems of Linear Equations and Inequalities (A.REI, N.VM)
QUARTER 2 – Discrete Math
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Chapter 12:
Chapter 13:
Chapter 14:
Chapter 15:
Probability (S.ID, S.CP)
Statistics (S.IC, S.ID)
Graph Theory (S.IC)
Voting and Apportionment (S.IC)
QUARTER 3 – Problem Solving
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Chapter 1: Critical Thinking Skills (G.MG, S. IC)
Chapter 9: Geometry (G.CO, G.SRT, G.CPE, G.MG)
Chapter 11: Consumer Mathematics (N.Q, N.RN)
Chapter 8: The Metric System (N.RN)
QUARTER 4 - Foundations
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Chapter 2: Sets (A.REI)
Chapter 3: Logic (G.CO)
Chapter 10: Mathematical Systems (N.RN)
Chapter 4: Systems of Numeration (N.RN)
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Number Theory and the Real Number System
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
To add, subtract, multiply and divide real numbers. Determining factorizations of real numbers using
divisibility rules. Predict the nth term and find the sum of arithmetic and geometric series and
sequences. Simplify radicals and expressions using rational exponents.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness; Health
Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 Common Core Curriculum Standards including Cumulative Progress Indicator (CPI):
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N-RN.2.
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
N-RN.3.
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number
and an irrational number is irrational; and that the product of a nonzero rational number and an
irrational number is irrational.
Unit Essential Questions:
What numbers are considered real numbers?
What operations can be used when working on real numbers?
Why is it important to be able to write numbers as the product of other numbers?
How do we classify various sets of numbers?
How can you use the properties of exponents to solve real world applications?
How do sequences and series model real world phenomena?
Unit Enduring Understandings:
Use properties of real numbers to solve real-life situations.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 perform real-number operations
 compute problems containing real numbers
 determining factorizations of real numbers using divisibility rules.
 calculate greatest common factor and least common multiple
 predict the nth term and find the sum of arithmetic and geometric series and sequences.
 simplify radicals and expressions using rational exponents.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessment
 Project
Formative Assessments:
 Homework Assignments
 Exit Ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
Use operations on real numbers to determine divisibility and factors, predict outcomes of sequences
and series, simplify fractions, radicals, and exponents, and translate verbal phrases into math symbols.
Hook the learner with engaging work.
Explore the Sieve of Eratosthenes to find all prime numbers less than or equal to a number n.
Challenge students to apply the algorithm to increasingly larger values of n.
Calculate the Golden Ratio using Fibonacci numbers. Explore the golden ratio’s importance in nature,
architecture and geometry.
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibInArt.html
Equip for understanding, experience and explore the big ideas.
Lesson on translating English phrases into math symbols and reinforce operations on real numbers.
Explore the Martingale System of wagering. (pg 274) Complete the associated problem (pg 277 #90).
Have students explain why they think this is a good strategy or a poor strategy.
Rethink opinions, revise ideas and work.
Reflect on homework problems and determine areas of concern.
The students will write their thoughts on this learning and then pair and share.
Evaluate your work and adjust as needed.
What questions and uncertainties do you still have in performing operations on real numbers,
translating verbal phrases, determining formulas for sequences and series?
Tailor the work to reflect individual needs, interests, and styles.
Assign classwork/homework over a range of abilities, allowing students to complete various problems
as they feel capable of completing.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.
 Begin with the hook.
 Introduce essential questions.
 Discuss uses of lesson and mathematical properties that apply.
 Direct instruction.
 Homework – Practice skills and short writing task.
 Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Algebra, Graphs, and Functions
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
In this unit the students will explore various representations of data and functions. They will construct a
scatter plot and line of best fit, identify a function and make an input-output table, identify when a
relation is a function and use the slope-intercept form or the standard form of a linear equation to graph
the equation.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness; Health
Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 New Jersey Core Curriculum Standards including Cumulative Progress Indicator (CPI):
A-REI 3
Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
F-IF 1, 2, 7a
1. Understand that a function from one set (called the domain) to another set (called the range) assigns
to each element of the domain exactly one element of the range. If f is a function and x is an element of
its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of
the equation y = f(x).
2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that
use fun
7. Graph functions expressed symbolically and show key features of the graph, by hand in simple
cases and using technology for more complicated cases.
a. Graph linear and quadratic functions and show intercepts, maxima, and minima. function notation in
terms of a context.
F-IF 3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of
the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n)
+ f(n-1) for n ≥ 1.
F-BF 1a, 1b
Write a function that describes a relationship between two quantities.
1a. Determine an explicit expression, a recursive process, or steps for calculation from a context.
1b. Combine standard function types using arithmetic operations. For example, build a function that
models the temperature of a cooling body by adding a constant function to a decaying exponential, and
relate these functions to the model.
from a table).
F-LE 5
Interpret the parameters in a linear or exponential function in terms of a context.
Unit Essential Questions:
What is a function?
What is functional notation?
What are the methods for graphing a linear equation?
How are functions used to display real-life situations?
What are scatter plots used for?
How can you determine what the line of best fit is?
Unit Enduring Understandings:
Functions in the form of a rule, an equation, a graph, table or a diagram can be used to transform
numbers into other numbers to solve problems.
Statistical measures and graphs can help you organize and make sense of data.
Visualize a linear equation and function by creating and analyzing its graphical representation.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 evaluate using function notation
 find domain and range of functions
 determine if a relation is a function.
 represent functions with tables, graphs, rules, and diagrams, including absolute value and piece
wise defined functions.
 make predictions given data from a function.
 evaluate functions
 students will be able to produce a graphical representation of a linear equation using data, two
points, point slope, and slope intercept form.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Systems of Linear Equations and Inequalities
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
In this unit, systems of equations and inequalities in two and three variables will be explored.
Students will identify and solve systems using the three methods of linear equations, determine the
number of solutions of a linear system and be able to graph and solve a system of linear inequalities.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness;
Health Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 New Jersey Core Curriculum Standards including Cumulative Progress Indicator (CPI):
A-CED 3
3. Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or nonviable options in a modeling context. For
example, represent inequalities describing nutritional and cost constraints on combinations of
different foods.
A-REI 1, 5, 6, 10, 12
1. Explain each step in solving a simple equation as following from the equality of numbers asserted
at the previous step, starting from the assumption that the original equation has a solution. Construct
a viable argument to justify a solution method.
5. Prove that, given a system of two equations in two variables, replacing one equation by the sum
of that equation and a multiple of the other produces a system with the same solutions.
6. Solve systems of linear equations exactly and approximately ( e.g., with graphs), focusing on pairs
of linear equations in two variables.
10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in
the coordinate plane, often forming a curve ( which could be a line)
12. Graph the solutions to a linear inequality in two variables as a half-plane (excluding the
boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes.
Unit Essential Questions:
What does the graph of two linear equations represent? What situation(s) would yield no solution?
What situation(s) would yield an infinite number of solutions?
What situation(s) would yield one solution and how does one find that solution?
What does the graph of two or more linear inequalities represent?
What is the difference between a bounded and non-bounded solution to a system of linear
inequalities?
Unit Enduring Understandings:
Real life situations can be represented algebraically using systems of equations.
The solutions are values that work for all equations and can be found graphically and algebraically.
Real life situations can be represented graphically using systems of inequalities.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 solve systems of equations.
 determine the solution(s) to systems of equations using substitution, graphing, and linear
combinations.
 identify key factors affecting the graphs of linear inequalities.
 graph systems of inequalities.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessments
 Projects
Formative Assessments:
 Demonstration
 Homework
 Class Discussion
 Exit Ticket
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher: Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
The students will extend knowledge of linear graph and inequalities to systems.
Hook the learner with engaging work.
Different Colors of Translucent paper: Modeling systems of inequalities using the overhead and the
colored sheets.
Question:
Suppose you have won a prize where you must spend exactly $1000 in 10 minutes on music CDs.
The store sells new releases for $10.00 and other CDs for $9.00 each. You want to know how many
of each type you need to buy if you want a total of 104 CDs. Explain that you can use a system of
linear equations to solve the problem.
Question:
Tell students that on a standardized test, 5 points are awarded for each correct answer and 1 point
is subtracted for each incorrect answer. Problems left unanswered are given a score of 0. If there
are 120 questions on the test and you get a score of 345, how many questions did you leave blank?
ACTIVITIES
Investigation: Graphs of Linear Systems
SWBAT: model a system of linear equations using the classroom as a coordinate plane.
Cop Chase
SWBAT: understand the limitations of solving systems graphically. The students will also be able to
solve systems of linear equations using substitution.
Escape from the Tomb Problem
SWBAT: Students are presented with a problem: two bowls are suspended from the ceiling by
springs. One bowl is lower than the other. In one bowl, you can only place marbles; in the other
bowl, you can only place bingo chips. The class should be divided into teams of three-four students.
One student is assigned the position as recorder. They will record the data from the experiments.
The second student is assigned the position as measurer. They will accurately measure the distance
from the bottom of the bowl to the floor (in centimeters). The third student is responsible for placing
items gently into the bowl. If bowls and springs are assembled and hung from the ceiling in advance,
each team will only need a tape measure, three copies of the activity packet, a calculator, a bag of
bingo chips, and a bag of marbles.
Questions to Ask:
How many items must be placed in each bowl so that the heights of the bowls are the same? If the
bowls and springs are not assembled in advance, each team will also need two bowls, two springs,
string and scissors. What can you tell me about this line? How do the slope and y-intercept relate to
the problem situation? What was the purpose of this activity? What did you learn? Was it what you
expected? Can you help Bart and Lisa solve their problem?
After the whole-class discussion, point to the two baskets that you have hanging in front of the room.
Give students the difference in height between these two baskets, and tell them that they have only
two minutes to figure out how many items should be placed in each basket. Allow students to begin
working, and while they are working, give an index card to each team. At the end of two minutes,
each team must write the names of their team members as well as their answer on the card. Then,
the cards should be given to you. One by one, allow the teams to come to the front of the room to
test their solutions. The team(s) with the most accurate answer can be given exact credit points or
some other reward.
Cell Phone Usage Comparison
SWBAT: write and graph equations to model allocation of money for cell phone usage for two
different companies. They will also be able to analyze the solution and the meaning of the graph.
Discuss who has cell phones in your class and what types of plans they have. You may want to
bring in current up to date cell phone plans to compare them. Tell students that they will be looking
at two prepaid plans offered by two different cell phone companies. Explain to students that their
parents have decided to buy them their first cell phone, and the parents have agreed to prepay $25
each month to be used for voice minutes and text messaging. Allow students time to look at the
chart and discuss which cell phone plan would be best under which circumstances before reading
through the questions on the activity sheet. Give groups time to complete Questions 1 to 5. Direct
students to graph their equations using whichever method they choose (slope and y-intercept, x/y
table, or x and y intercepts). You may also want to discuss the meaning of negative x- and y-values.
Questions to ask:
Is it possible to send a negative number of text messages or talk on the phone for negative minutes?
Students should realize that only positive values make sense in this problem, and therefore they
should be graphing in quadrant I only. Did they choose the plan they thought they would before
working through the activity? Which plan did they choose and why? Which plan was most popular in
the class?
Equip for understanding, experience and explore the big ideas.
Many problems lend themselves to being solved with systems of linear equations. In "real life", these
problems can be incredibly complex. This is one reason why linear algebra (the study of linear
systems and related concepts) is its own branch of mathematics. In your studies, however, you
should generally be faced with much simpler problems. What follows are some typical examples.
The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200
people enter the fair and $5050 is collected. How many children and how many adults attended?
In the past I would have set this up by picking a variable for one of the groups (say, "c" for "children")
and then use "(total) less (what I've already accounted for)" (in this case, "2200 – c") for the other
group. Using a system of equations, however, allows me to use two different variables for the two
different unknowns.
Solve this word problem using a system of equations.
Rethink opinions, revise ideas and work.
Discuss your finding with a peer and journal any concerns or interesting findings.
Reflect on homework problems and determine areas of concern.
The students will write their thoughts on this learning activity and then pair and share.
Evaluate your work and adjust as needed.
Students will be given a problem to try individually. At this time, the student will be able to assess
his/her own progress. The student will be given time in class to ask any additional questions to
clarify their understanding.
What questions and uncertainties do you still have in this lesson?
Discuss this with your group.
Tailor the work to reflect individual needs, interests, and styles.
Assign class work/homework over a range of abilities, allowing the students to complete various
problems as they feel capable of completing.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.
 Begin with the hook.
 Introduce essential questions.
 Discuss uses of lesson and mathematical properties that apply.
 Direct Instruction.
 Homework – Practice skills and short writing task.
 Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Probability
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
This unit will help develop the ability to count the number of ways an event can happen and to
calculate and use probabilities and odds. This unit will also focus on problems in life that can be
solved using probability, combinations and permutations.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness; Health
Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 Common Core Curriculum Standards including Cumulative Progress Indicator (CPI):
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S-CP.1.
Describe events as subsets of a sample space (the set of outcomes) using characteristics (or
categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,”
“not”).
S-CP.2.
Understand that two events A and B are independent if the probability of A and B occurring together is
the product of their probabilities, and use this characterization to determine if they are independent.
S-CP.3.
Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of
A and B as saying that the conditional probability of A given B is the same as the probability of A, and
the conditional probability of B given A is the same as the probability of B.
S-CP.6.
Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and
interpret the answer in terms of the model.
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S-CP.7.
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of
the model.
S-CP.8.
(+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) =
P(B)P(A|B), and interpret the answer in terms of the model.
S-CP.9.
(+) Use permutations and combinations to compute probabilities of compound events and solve
problems.
Unit Essential Questions:
How do you count the ways an event can happen?
What is luck?
How can we use probabilities and expected values to influence our decision making?
Unit Enduring Understandings:
The difference between empirical and theoretical probability
The concepts of sample space, events, outcomes, unions, intersections and complements.
The definition of independent and dependent events.
The definition of conditional probability.
When it is appropriate to use permutations and combinations
The addition and general multiplication rules, and their uses.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 Describe the union and intersection of an events, and the complement of an event.
 Justify that two events are independent.
 Determine the conditional probability of an event given another event has already occurred.
 Compute the probability of A or B
 Compute the probability of A and B
 Compute probabilities of compound events using permutations and combinations.
 Apply the binomial probability formula.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessment
 Project
Formative Assessments:
 Homework Assignments
 Exit Ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:



Collins Writing Tasks
Notebook
Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
Use various techniques to calculate probability. Differentiate between, and select from a variety of
methods to solve problems involving probability. Understand where probabilities and odds apply in the
real world and use these examples to contrast empirical and theoretical probability.
Hook the learner with engaging work.
Discuss the rate of occurrence for various genetic disorders. (pg 677) How did scientists arrive at
these numbers? Did they test exactly 2500 people? More or less?
Calculate the expected value of several carnival and / or casino games. Is it worth it to play one game
as opposed to another? What expected value would make a game a good one to play? A bad one?
Equip for understanding, experience and explore the big ideas.
When is probability used?
Discuss the Birthday Problem (pg721). Are there any other events with probabilites that are
counterintuitive?
Rethink opinions, revise ideas and work.
Reflect on homework problems and determine areas of concern.
Think of when the concepts discussed in the chapter have had an effect on your life.
Evaluate your work and adjust as needed.
What questions and uncertainties do you still have in calculating probabilites, odds, and expected
value? Are the concepts of independent and dependent events still unclear?
Students will create their own examples that meet selected criteria. Classmates will determine if
examples created are appropriate and will assist in refining ideas.
Tailor the work to reflect individual needs, interests, and styles.
Assign classwork/homework over a range of abilities, allowing students to complete various problems
as they feel capable of completing.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.






Begin with the hook.
Introduce essential questions.
Discuss uses of lesson and mathematical properties that apply.
Direct instruction.
Homework – Practice skills and short writing task.
Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Statistics
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
In this unit measures of central tendency and measures of dispersion will be used to describe sets of
data. Discussions and examples of the misuse of statistics will be used to make more accurate
interpretations. This unit will focus on using statistics and statistical graphs to analyze real-world
situations.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness; Health
Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 Common Core Curriculum Standards including Cumulative Progress Indicator (CPI):












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S-ID.1.
Represent data with plots on the real number line (dot plots, histograms, and box plots).
S-ID.2.
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and
spread (interquartile range, standard deviation) of two or more different data sets.
S-ID.3.
Interpret differences in shape, center, and spread in the context of the data sets, accounting for
possible effects of extreme data points (outliers).
S-ID.4.
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate
population percentages. Recognize that there are data sets for which such a procedure is not
appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
S-ID.8.
Compute (using technology) and interpret the correlation coefficient of a linear fit.
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S-ID.9.
Distinguish between correlation and causation
Unit Essential Questions:
Why is data collected and organized?
How can data be used to influence people?
How can predictions be made based on data?
How can data be visually represented?
Unit Enduring Understandings:
Understand the meaning of random sample, statistic, parameter, population and sample.
Use the components of a survey, an experiment and an observational study.
Know the different types of statistical graphs and their uses.
Know the different measures of central tendency and their uses.
Understand the measures of dispersion and their uses.
Know the properties of the normal distribution.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 make inferences about a population from a random sample.
 determine if statistics are represented accurately.
 create a histogram, stem and leaf plot, and circle graph
 interpret data represented in a histogram, stem and leaf plot or circle graph
 calculate measures of central tendency
 calculate dispersion
 apply the normal distribution to real world problems
 calculate the z-score of an item from a data set.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessment
 Project
Formative Assessments:
 Homework Assignments
 Exit Ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
The student will be able to use measures of central tendency and measures of dispersion to describe
sets of data.
The student will be able to use statistics and statistical graphs to analyze real-world situations.
Hook the learner with engaging work.
The student can use a graphing calculator to find statistics and draw statistical graphs.
EXAMPLE.
The fat content and number of calories in several different sandwiches available at a restaurant are
shown in the tables below. Use a graphing calculator to:
(a) find the mean , median, mode, and range of the fat content in the sandwiches,
(b) draw a box-and-whisker plot of the fat content in the sandwiches, and
(c) draw a histogram of the number of calories in the sandwiches.
Sandwich
Hamburger
Cheeseburger
Quarter-pound hamburger
Quarter-pound cheeseburger
Double cheeseburger
Bacon cheeseburger
Fried chicken
Grilled chicken
Breaded fish on deluxe roll
Breaded fish on plain bun
Fat (g)
9
13
21
30
31
34
25
20
28
25
Calories
260
320
420
530
560
590
500
440
560
450
SOLUTION.
a) Use the Stat Edit feature to enter the data in a list. Then use the Stat Calc menu to choose 1variable statistics. The mean is x-bar and by scrolling you can find the median (Med). The
range is the difference of maxX and minX .
b) Use the Stat Plot menu to choose the type of plot (box-and-whisker), the list of data, and the
frequency for the data. Then set an appropriate viewing window. Draw the box-and-whisker.
Use the Trace feature to view the minimum (9), the lower quartile (Q1/20), the median (25), the
upper or third quartile (Q3/30), and the maximum (34).
Equip for understanding, experience and explore the big ideas.
Extend the hook and use statistics and statistical graphs to analyze real-life data sets, such as the freethrow percentages for the players in the NBA.
Have students graph and discuss the results using their graphing calculators.
Rethink opinions, revise ideas and work.
Have open discussions on putting the information in the graphing calculator and interpreting the results
using the mean, mode, median, and range.
Evaluate your work and adjust as needed.
What questions and uncertainties do the students still have about using the graphing calculator to input
information and draw a box-and-whisker graph.
Tailor the work to reflect individual needs, interests, and styles.
Have the students model real-life problems that use measures of central tendency and dispersion.
Have students use standard deviation as another measure of dispersion.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.
 Begin with the hook.
 Introduce essential questions.
 Discuss uses of lesson and mathematical properties that apply.
 Direct instruction.
 Homework – Practice skills and short writing task.
 Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Graph Theory
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
In this unit, students will learn the vocabulary and terms necessary to model scenarios using weighted
and unweighted graphs. Graphs will be analyzed with Euler and Hamiltonian paths and circuits.
Classical problems like the Traveling Salesman and Seven Bridges of Konigsberg will be used to
introduce lessons and test algorithms.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness; Health
Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 Common Core Curriculum Standards including Cumulative Progress Indicator (CPI):
S-CP.9.
(+) Use permutations and combinations to compute probabilities of compound events and solve
problems.
G-MG.1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree
trunk or a human torso as a cylinder)
G-MG.3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy
physical constraints or minimize cost; working with typographic grid systems based on ratios)
Unit Essential Questions:
Why do we organize information in graphs?
Why is it useful to know if a graph has an Euler path or circuit?
Why is it useful to know if a graph has a Hamiltonian path or circuit?
What are the methods to find minimum distance paths and circuits?
How do we determine the minimum-cost spanning tree from a weighted graph?
Unit Enduring Understandings:
Know the terms: weighted, unweighted, path, circuit, Euler, Hamiltonian.
Understand that Euler paths and circuits are an analysis of the edges of a graph, while Hamiltonian
paths and circuits are an analysis of the vertices of a graph.
Know how even and odd vertices determine the paths and circuits of a graph.
Understand that an algorithm is a process to follow to reach a desired outcome.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 draw weighted and unweighted graphs to represent real scenarios
 determine if a graph has an Euler path or circuit using Fleury’s Algorithm.
 determine if a graph has a Hamiltonian path or circuit.
 determine the number of unique Hamiltonian circuits in a complete graph.
 calculate approximate solutions to Traveling Salesman problems.
 use Kruskal’s Algorithm to construct the minimum-cost spanning tree from a weighted graph.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessment
 Project
Formative Assessments:
 Homework Assignments
 Exit Ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
The student will be able to represent real scenarios using weighted and unweighted graphs.
The student will use various algorithms to analyze weighted and unweighted graphs to make decisions
about real world problems.
Hook the learner with engaging work.
Introduce the Seven Bridges of Konigsberg problem. (pg. 854) Can we represent the map in a more
simplified way to make it easier to analyze? Does having the graph without the unecessary details
make it easier to trace paths around the graph? Are you convinced that the problem is solvable or
unsolvable? What information would you like to see to make an informed decision?
Facebook Project: Have students create a poster depicting a graph of three degrees of facebook
friends. Analyze who is connected to who. How many celebrities or sports figures are connected to
students in the class?
Traveling Salesman: Create your own traveling salesman problem with towns and cities in the
Philadelphia / South Jersey Area. Use printouts from Google maps to highlight the paths and to find
accurate mileage between vertices.
Equip for understanding, experience and explore the big ideas.
Extend the hook to planning a trip around the world, being sure to visit sites agreed upon by the class.
Use real pricing data and create a class graph of the trip. Does our graph have any Euler / Hamiltonian
paths or circuits? Can we minimize the cost of our trip?
Rethink opinions, revise ideas and work.
Assign regions of the world for individual students to transform into graphs and analyze for Euler or
Hamiltonian circuits / paths. Students should check each others workd and discuss any errors or
difficulties that have arisen.
Evaluate your work and adjust as needed.
After creating and analyzing several weighted and unweighted graphs, are there any concepts that are
still unclear? How could you help a classmate that is struggling with analysis? Create a graphic
organizer, table, or other study aid to assist in memorizing important terms and concepts.
Tailor the work to reflect individual needs, interests, and styles.
Allow students to create and explore graphs from disciplines that interest them. (Home / Business
Computer networks, Road Trips, Social Networking)
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.
 Begin with the hook.
 Introduce essential questions.
 Discuss uses of lesson and mathematical properties that apply.
 Direct instruction.
 Homework – Practice skills and short writing task.
 Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Voting and Apportionment
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
The students will learn how the various voting and apportionment methods are used and what their
flaws are.
21st Century Skills: Critical thinking and problem solving; Communication; Collaboration; Creativity
and Innovation
21st Century Themes: Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy;
Global Awareness; Health Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 New Jersey Core Curriculum Standards including Cumulative Progress Indicator (CPI):





S-IC.2.
Decide if a specified model is consistent with results from a given data-generating process, e.g.,
using simulation. For example, a model says a spinning coin falls heads up with probability 0.5.
Would a result of 5 tails in a row cause you to question the model?
S-ID.4.
Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate
population percentages. Recognize that there are data sets for which such a procedure is not
appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
Unit Essential Questions:
How are elections run and how are discrete sets apportioned?
What are the advantages and disadvantages of different methods?
Unit Enduring Understandings:
There are different ways to find the winner of an election and to split up an estate. Some are more
appropriate than others in certain situations.
Use the election and apportionment algorithms correctly.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 Use various voting methods correctly.
 Determine the weaknesses of the methods.
 Use various apportionment algorithms correctly.
 Determine the weaknesses of the methods.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessments
 Projects
Formative Assessments:
 Homework Assignments
 Exit Ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
Having the ability to use any algorithm is a valuable skill in life. Being able to use the algorithms
presented in this unit will help students get better at using algorithms. Then, being able to evaluate
the algorithms engages higher lever thinking skills.
Hook the learner with engaging work.
The first lesson on voting methods should involve students voting for something they are interested
in, for example favorite soft drinks. Students will vote in the election for their favorite soft drinks.
Different groups should get together and evaluate who they feel is the winner of the election. At the
end of lesson the students can present their findings to the class. And other groups can then
evaluate the group on their method. when actual voting methods are learned, the students can use
the real method for finding real winners in their soft drink election.
The first lesson on apportionment algorithms can involve a group of students dividing up an estate.
Allow the students to use their own ideas for coming up with a fair way to divide the items that are in
An estate amongst the heirs. And then after the students learn a real algorithm, they can find the
real fair division of the estate.
Title Fair is Fair (Greta Lawlor)
Goals 1) Students will explore several voting methods and decide which is the "most" fair
using Arrow's Conditions.
2) Students will be able to use and justify a method for determining a group ranking and apply
Arrow's Method to justify that their method is fair.
3) Students will work in small groups to come to a consensual solution and be able to justify
that solution.
Abstract This activity is set in the context of choosing a site for a recreation center geared at
teenagers. Students are to determine a group ranking based on a set of already collected
preferences and to apply several voting methods to decide on the site for the recreation
center. They will use Arrow's conditions to justify the "fairness" of their solution.
Problem Statement Discuss with students what advantages the recreation center might have
for a community. Also, discuss the disadvantages. Let students know that they will be looking
at the problem in the context that the recreation center is sought after by all communities
under consideration. The students will be examining the voter information and examining
ways to some up with a fair way to decide the location of the center.
Instructor Suggestions 1) Set the stage with the discussions of the advantages and
disadvantages of the center.
2) Distribute the activity "Fair is Fair" and after the students have read the problem, discuss
any questions they may have.
3) Have the students work in groups and use at least two methods to determine a winner for
the recreation center.
4) Have the students present their solutions and justify their solution using Arrow's conditions.
5) Discuss the students work in relation to "fairness."
Equip for understanding, experience and explore the big ideas.
Student groups will run an actual election in their homerooms on a designated day. They will then
find the results of the election and present a winner. Different groups will run elections on different
things. The results of each collection will be presented to the entire school through a news
broadcast.
Rethink opinions, revise ideas and work.
Students will compare and contrast the strengths and weaknesses of various election and
apportionment algorithms. Students will also be able to critique their own methods.
Evaluate your work and adjust as needed.
Do you think there are still better algorithms out there? Do you think it's important for people to study
these algorithms to come up with improved ones?
Tailor the work to reflect individual needs, interests, and styles.
Assign homework and classwork over a range of abilities. Allow the students to complete various
problems as they feel capable. Allow higher-level students to work extra projects as needed.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.






Begin with the hook.
Introduce essential questions.
Discuss uses of lesson and mathematical properties that apply.
Direct Instruction.
Homework – Practice skills and short writing task.
Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Critical Thinking Skills
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
The students will learn to use inductive reasoning, estimation, and problem solving skills in order to
complete problems described using words.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness;
Health Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 New Jersey Core Curriculum Standards including Cumulative Progress Indicator (CPI):




G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure
to satisfy physical constraints or minimize cost; working with typographic grid systems based on
ratios).★
S-IC.2. Decide if a specified model is consistent with results from a given data-generating process,
e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5.
Would a result of 5 tails in a row cause you to question the model?
Unit Essential Questions:
What techniques can I use to read through the words in an application problem in order to get to the
math?
Unit Enduring Understandings:
Rewrite the important information in a problem without all of the words.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 outline and solve word problems.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessments
 Projects
Formative Assessments:
 Homework Assignments
 Exit Ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
Students need to realize that working with word problems not as difficult as they think it is. Using a
technique such as outlining the important mathematical parts of the problem will help them to
become more proficient at solving word problems. Students should use inductive reasoning,
estimation, and problem solving skills to tackle these application problems.
Hook the learner with engaging work.
Have the students work in reverse by taking a numerical mathematical problem, and changing it to a
word problem. Groups of students can exchange word problems and discuss various ways of solving
the mathematical portion of that problem.
http://www.yourteacher.com/geometry/inductivereasoning.php
Equip for understanding, experience and explore the big ideas.
Give student real problems from real sources such as news broadcasts newspapers and magazines.
Have students solve these problems correctly.
Rethink opinions, revise ideas and work.
After learning an outlining technique for word problems students can then revert back to an old
method by trying to make calculations in their head and see that the outline method works much
better.
Evaluate your work and adjust as needed.
How has your ability to solve a word problem improved after learning how to use an outlining
technique?
Tailor the work to reflect individual needs, interests, and styles.
Assign homework and classwork over a range of abilities. Allow the students to complete various
problems as they feel capable. Allow higher-level students to work extra projects as needed.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.
 Begin with the hook.
 Introduce essential questions.
 Discuss uses of lesson and mathematical properties that apply.
 Direct Instruction.
 Homework – Practice skills and short writing task.
 Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Geometry
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
In this unit, students will extend upon several important concepts from Geometry, including Lines and
Angles, Polygons, Similarity, Volume, Surface Area, Transformations and the Pythagorean Theorem.
Students will also get a chance to explore an introduction to the field of Topology.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness; Health
Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 Common Core Curriculum Standards including Cumulative Progress Indicator (CPI):













G.CO.1.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on
the undefined notions of point, line, distance along a line, and distance around a circular arc.
G-CO.2.
Represent transformations in the plane using, e.g., transparencies and geometry software; describe
transformations as functions that take points in the plane as inputs and give other points as outputs.
Compare transformations that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
G-CO.3.
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections
that carry it onto itself.
G-CO.4.
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular
lines, parallel lines, and line segments.


G-CO.5.
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using,
e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will
carry a given figure onto another.
G-MG.1.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree
trunk or a human torso as a cylinder)
G-MG.3.
Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy
physical constraints or minimize cost; working with typographic grid systems based on ratios)



G-SRT.2.
Given two figures, use the definition of similarity in terms of similarity transformations to decide if they
are similar; explain using similarity transformations the meaning of similarity for triangles as the equality
of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G-GMD.3.
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Unit Essential Questions:
Why are some terms undefined?
How can points, lines and planes interact to form the geometric figures we know? What are their
properties?
What formulas are used in Geometry? When is it appropriate to use each formula?
What does it mean to be similar? How can similarity be used to solve problems?
What is Topology?
Unit Enduring Understandings:
Know properties of Geometric figures.
Understand reflections, rotations and translations result in congruent figures, while dilations do not.
Know that there are various methods to calculate missing segments and angles in figures.
Know that an algorithm is a process to follow to reach a desired outcome.
Know that figures like the Mobius Strip, Klein Bottle and Jordan Curve have properties unlike the
figures we’ve studied so far in Geometry.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 calculate missing angles and segments in various Geometric figures.
 determine if figures are similar, and use the properties of similarity to find missing values.
 calculate perimeter, area, volume and surface area of Geometric figures.
 describe and perform transformation on an object in the coordinate plane.
 create a tessellation.
 describe the topological properties of figures (Mobius Strip, Klein Bottle).
 describe the genus of objects as it pertains to topology.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessment
 Project
Formative Assessments:
 Homework Assignments
 Exit Ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
Students will reinforce the concepts that they have studied in their formal Geometry courses. Topology
and fractals will be discussed to broaden students’ Geometric knowledge.
Hook the learner with engaging work.
Introduction to Unit: Many objects that we encounter can be described in terms of Geometry. Look
around the room, or think of items you encounter in your daily life. Try to describe them in terms of
Geometry. Using each students’ list, create a classroom comparison table of all of the real world
objects with their geometric counterpart.
Indirect Measurement: Have students go outside to measure the height of the school, light posts,
flagpoles, etc. Using the shadows of figures and known heights, set up similar triangles and calculate
measurements.
Tesselations: Create a translation tessellation using Geometer’s Sketchpad
Topology and Fractals: Use the internet to explore interactive versions of the Klein Bottle and Fractals.
Have students create their own pattern to extend in Fractal form.
Equip for understanding, experience and explore the big ideas.
Formal Constructions: Explore construction techniques used for creating perpendicular lines, including
constructing a perpendicular bisector and constructing a perpendicular segment through a given point
on a given line.
Proving Quadrilateral Characteristics. Using Geometer's Sketchpad, students construct a
parallelogram, rectangle, rhombus and square. Students then use the measurement tool and complete
graphic organizer to prove each quadrilateral by both definition and specific characteristic
Rethink opinions, revise ideas and work.
What concepts do you remember from Geometry? What was difficult? What was easy? What ideas
would you like to spend more time on?
Evaluate your work and adjust as needed.
Review your classwork and homework? Are there any graphic organizers that would help you
remember important properties?
Tailor the work to reflect individual needs, interests, and styles.
Are there any other methods that you know to find heights of very tall objects? (Trigonometry)
Explore the history of the Pythagorean Theorem and its proofs. Allow students to go online and find
graphical and algebraic proofs of the theorem.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.
 Begin with the hook.
 Introduce essential questions.
 Discuss uses of lesson and mathematical properties that apply.
 Direct instruction.
 Homework – Practice skills and short writing task.
 Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Consumer Mathematics
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
Students will discover the mathematics that are used everyday by consumers everywhere. They will
use percents, interest formulas, mortgage formulas, and investment formulas.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness; Health
Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 New Jersey Core Curriculum Standards including Cumulative Progress Indicator (CPI):



N-RN.2.
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
N-Q.1.
Use units as a way to understand problems and to guide the solution of multi-step problems; choose
and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and
data displays


o
o
o
o
o
F-IF.7.
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases
and using technology for more complicated cases.★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute
value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing
end behavior.
d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are
available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric
functions, showing period, midline, and amplitude.
Unit Essential Questions:
How do ordinary people use mathematics everyday?
Unit Enduring Understandings:
It is important to develop mathematical skills to be able to check computations and not get ripped off.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 Confirm bills produced by professionals and find mistakes when made.
 Also discover who benefits from mistakes.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessments
 Projects
Formative Assessments:
 Homework Assignments
 Exit ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
Students will come to an understanding that being able to use basic percent, simple interest, compound
interest, and mortgage retirement investment formulas will be advantageous to them later in life.
Hook the learner with engaging work.
Students will take a consumer mathematics problem home in order to have their parents work the
problem with the students. Students will have discussions with parents as to what types of consumer
mathematics problems the parents solve every day. Students will hopefully find out that the ability to
solve these problems will help them in adult life.
On the million dollar project.
http://www.proteacher.net/discussion...ead.php?t=7568
Equip for understanding, experience and explore the big ideas.
Students will exchange the consumer mathematics problem that they took home for their parents, with
each other. They will work the problem groups and solve it correctly.
Rethink opinions, revise ideas and work.
Students will analyze different situations where professionals make mistakes and charge consumers too
much money. Then they will come up with an interpretation of what the impact will be in the
consumer's life.
Evaluate your work and adjust as needed.
Do you think after studying this unit you will be a better consumer? If not, what topics do you need to
study more?
Tailor the work to reflect individual needs, interests, and styles.
Assign homework and classwork over a range of abilities. Allow the students to complete various
problems as they feel capable. Allow higher-level students to work extra projects as needed.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.
 Begin with the hook.
 Introduce essential questions.
 Discuss uses of lesson and mathematical properties that apply.
 Direct Instruction.
 Homework – Practice skills and short writing task.
 Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
The Metric System
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
This unit is devoted to the understanding and applying the metric system. Students will work towards
having a better understanding of the size of certain measurements and appropriate use of units for
measuring objects. Students will also convert among units in the metric system and between metric
and customary units.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness; Health
Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 Common Core Curriculum Standards including Cumulative Progress Indicator (CPI):



N-RN.3.
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number
and an irrational number is irrational; and that the product of a nonzero rational number and an
irrational number is irrational.

Unit Essential Questions:
What are the basic units of the metric system?
How do we convert between metric units?
How do we convert metric units to other measurement systems?
How can we choose appropriate units for our measurements?
Unit Enduring Understandings:
Know the basic units for length, mass, volume, temperature.
Know the prefixes for converting within the metric system.
Understand the relative sizes of metric units.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 convert among the units in the metric system.
 convert metric measurements to customary measurements using dimensional analysis.
 convert customary measurements to metric measurements using dimensional analysis.
 choose appropriate units when measuring objects.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessment
 Project
Formative Assessments:
 Homework Assignments
 Exit Ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
The student will be able to convert measurements to units within the metric system and customary
units. The unit is designed to give the students a ‘feel’ for the size of measurements and the
appropriate use. We would not use grams to measure a football player, nor would we use square
centimeters to measure the size of a soccer pitch.
Hook the learner with engaging work.
Ask students to estimate the length, mass, volume, etc. of different objects. Discuss appropriate units
and if the units chosen make sense.
Gatorade is poured into a plastic bottle that holds 1.2 liters of liquid. The bottle is then placed in a
freezer. When the bottle is removed from the freezer, the plastic is cut away, leaving just the frozen
Gatorade. What is the approximate mass of the frozen Gatorade in grams? What is the approximate
volume of the frozen Gatorade in cubic centimeters?
The displacement of automobile engines is measured in liters. Research the stock displacement for at
least five vehicles and determine the displacements in cubic centimeters and cubic inches.
Equip for understanding, experience and explore the big ideas.
Students can research conversion factors for US Dollars to other currencies and develop unit fractions
for different conversions.
The importance of the metric system for anyone interested in medical professions cannot be
understated. The following problems involve applications of the metric system to medicine.
1. Twenty milligrams of the drug lincomycin is to be given for each kilogram of a person’s
weight. The drug is to be mixed with 250 cc of a normal saline solution, and the mixture is to be
administered intravenously over a 1-hr period. Clyde Dexter, who weighs 196lb, is to be given
the drug. Determine the dosage of the drug he will be given.
At what rate per minute should the 250 cc solution be administered?
2. At a pharmacy, a parent asks a pharmacist why her child needs such a small dosage of a
certain medicine. The pharmacist explains that a general formula may be used to estimate a
child’s dosage of certain medicines. The formula is
𝑐ℎ𝑖𝑙𝑑𝑠 𝑤𝑒𝑖𝑔ℎ𝑡 𝑖𝑛 𝑘𝑔
× 𝑎𝑑𝑢𝑙𝑡 𝑑𝑜𝑠𝑒
67.5𝑘𝑔
What is the amount of medicine you would give a 60 lb child if the adult dose is 70mg?
𝐶ℎ𝑖𝑙𝑑′ 𝑠 𝑑𝑜𝑠𝑒 =
At what weight (in pounds), would the child receive an adult dose?
Rethink opinions, revise ideas and work.
The metric system is discussed at length in your Science courses. What information did you remember
from those classes? Is there anything that is still unclear? Was any information clarified for you by
looking at it in a different light in a Math class?
Evaluate your work and adjust as needed.
Allow students to look for a family recipe and convert all measurements to metric. Students will then
be placed in groups, trade recipes, and verify if the conversions are correct.
Use Math journals to allow students to do daily conversion warm ups. Students can then trade journals
to check the accuracy of their peers’ results.
Tailor the work to reflect individual needs, interests, and styles.
You are asked to create an I-phone / I-pad app that will do conversions if a base unit and conversion
unit are selected. What conversion factors would need to be included? Would there be a way to
reduce the number of conversion factors that would be programmed into the app?
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.
 Begin with the hook.





Introduce essential questions.
Discuss uses of lesson and mathematical properties that apply.
Direct instruction.
Homework – Practice skills and short writing task.
Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Sets
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
In this unit, students will learn how things are sorted and classified using mathematical notation.
Venn Diagrams will be used to visually represent interactions between sets. Operations such as
complement, intersection, union, difference and Cartesian product will be examined. Real world
applications of sets and problems involving sets will be discussed.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness;
Health Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 Common Core Curriculum Standards including Cumulative Progress Indicator (CPI):








o
F-IF.1.
Understand that a function from one set (called the domain) to another set (called the range) assigns
to each element of the domain exactly one element of the range. If f is a function and x is an element
of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the
graph of the equation y = f(x).
S-ID.3.
Interpret differences in shape, center, and spread in the context of the data sets, accounting for
possible effects of extreme data points (outliers).
S-ID.6.
Represent data on two quantitative variables on a scatter plot, and describe how the variables are
related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data.
Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and
exponential models.
Unit Essential Questions:
Why do we organize information in sets?
How can Venn diagrams be used to show the interaction among sets?
How can sets be subdivided into smaller sets?
What operations can be performed on sets?
How can sets be used to help us solve problems?
Unit Enduring Understandings:
Know the terms associated with set theory: Roster Form, Set-Builder Notation, Subset, Proper
Subset, Complement, Intersection, Union, Difference, Cartesian Product, Cardinality, Infinite Set.
Understand that sets can be described using roster form or set-builder notation.
Know that Venn Diagrams allow us to visualize sets and their interactions.
Know the symbols used in set theory: ′, ⋃ , ⋂ , −,×, ∈,
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 describe sets using Set-Builder Notation, and Roster Form
 determine the cardinality of a set, if an item is an element of a set, if a set is empty.
 create Venn diagrams describing the interaction between two and three sets.
 interpret Venn diagrams to build sets of items.
 perform the operations Union, Intersection and Complement on sets.
 determine the number of elements in sets joined by ‘and’.
 compute the Cartesian product of two sets.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessment
 Project
Formative Assessments:
 Homework Assignments
 Exit Ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad




Rulers / meter sticks
White Board / Graph Board
Graphing Calculators
Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
The students will understand the terminology and notations that allow us to sort and classify items in
sets. They will perform various operations on sets and examine the interaction of two or more sets.
Students will have the opportunity to use sets to classify and organize objects to assist in solving
problems.
Hook the learner with engaging work.
Chapter Introduction: How would you classify a McDonalds restaurant? What categories or groups
does it belong to? (Fast food, Hamburger Restaurant, Breakfast restaurant). Decide on one
classification and then list other places that would fall into the same set. Which restaurants would be
included in some classifications, but excluded from others. (For example, if we classify McDonalds
as fast food, then Taco Bell would be included in the set. However, if we classify it as a restaurant
that sells hamburgers, Taco Bell would be excluded.)
Venn Diagrams: Give students two criteria (Such as a 15’’ laptop display and an 8gb hard drive).
Allow them to search the internet for laptops that have either a 15” display or an 8gb hard drive and
laptops that have both. Organize the data into a Venn diagram. Discuss how adding more criteria
would alter the diagrams. Can we think of properties that would never intersect in the Venn diagram?
Create a Venn diagram with three criteria by surveying the class. Make a list of students who are
involved in sports, clubs, non school funded activities. Organize the data into a Venn diagram and
discuss the results.
Equip for understanding, experience and explore the big ideas.
Have students create a 5-10 element set from a list of given categories. Experiment with the union,
intersection, complement etc of sets. See how many sets the class can effectively handle at one
time. Did anyone create equivalent sets?
Rethink opinions, revise ideas and work.
Review projects and assignments. Could any of the projects be done with a different category for the
sets? How would the outcomes change?
Evaluate your work and adjust as needed.
What did you learn? Was there any information that was common sense to you or already known
before beginning this unit? Are any concepts still unclear? How could you easily describe an infinite
set?
Tailor the work to reflect individual needs, interests, and styles.
Allow students to create and explore sets from disciplines that interest them.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.

Begin with the hook.





Introduce essential questions.
Discuss uses of lesson and mathematical properties that apply.
Direct instruction.
Homework – Practice skills and short writing task.
Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Logic
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
Students will build truth tables, make equivalent statements and symbolic arguments, and draw
Euler diagrams and switching circuits.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness;
Health Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 New Jersey Core Curriculum Standards including Cumulative Progress Indicator (CPI):







G.CO.1.
Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on
the undefined notions of point, line, distance along a line, and distance around a circular arc.
S-ID.9.
Distinguish between correlation and causation.
S-IC.6.
Evaluate reports based on data.
Unit Essential Questions:
How do we make logical statements and connectives?
Why do we build a truth table?
What makes statements equivalent?
How do we make symbolic arguments?
What type of problems can be solved using Euler diagrams or switching circuits?
Unit Enduring Understandings:
Computers are logical mathematical creatures. Logic is the basis for everything about a computer.
Learn logic to be able to learn computers.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 make logical statements using correct notation.
 build truth tables and use them to make conditional statements.
 decide if statements are equivalent.
 make symbolic arguments.
 build an Euler diagram and interpret it correctly.
 build a switching circuit to make a decision.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessments
 Projects
Formative Assessments:
 Demonstration
 Homework
 Class Discussion
 Exit Ticket
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher: Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
Students will come to an understanding that logic is the basis for all computer languages and
computer thinking. Being able to make logical statements, truth tables, conditional and biconditional
statements, equivalent statements, symbolic arguments, Euler diagrams and switching circuits is the
basis for thinking logically.
Hook the learner with engaging work.
Students can be placed into groups and asked to come up with a number of different ways an event
can occur without using any sort of pencil, paper, or calculator. Then students can discuss amongst
their groups to see who was able to come up with the correct values. Finally students can organize
their work into a truth table to manage the data and solve the problem.
Students can play the logic game as outlined nicely on page 168 in the text.
Help students distinguish between believing a claim and supporting it. Start with a claim that most of
your students won’t accept. I like to use, “Students shouldn’t be allowed to own cars.” I acknowledge
that we all disagree with the statement, and then I offer some supporting statements:



“Students have more accidents than older drivers.” I ask a few students to search for
evidence of this claim on the classroom computers or their smartphones.
“The costs of owning a car make students work more, giving them less time to study.” Again,
we search for facts and figures to back this up.
“Not having cars encourages students to spend more time on campus or with other students,
increasing their involvement with and commitment to school life.” For this one, we try polling
class members, and discuss when we can rely on our own experience or that of people we
know, and when we need a larger sample size.
Have students choose (or choose for the class) current events topics. Give time to research the
issues, and ask students to list all the different points of view they find for the topic in the course of
their research, and list the arguments given in support of those points of view. I use this as a
preliminary for exercises on response to a text, so we work on presenting the arguments fairly,
regardless of our opinions of them, and on critical analysis of the points made.
Hold a simple debate. Choose a topic that is familiar enough to students that they’ll be able to
understand it easily and think of arguments on both sides of it, but which won’t be so emotional that
students will get distracted. A local controversy, vampires vs. werewolves, should human cloning be
allowed… You know your population best, so you can choose. Divide the class randomly into two
groups and assign each one side of the argument. Allow the teams time to prepare their support for
the side they’ve been assigned. Let the teams take turns presenting their support for ten minutes.
After ten minutes, the two teams should switch sides. The team that presents the largest number of
valid arguments wins.
Equip for understanding, experience and explore the big ideas.
Students will research occupations where logic plays a vital roll. And present their findings in the
form of a project.
Rethink opinions, revise ideas and work.
After working homework problems students can check their answers and be prepared to ask
questions in class about anything they did not understand.
Evaluate your work and adjust as needed.
Students will work in groups to assess each other's knowledge of unit topics. Anything that is not
completely understood can be reviewed and relearned through the use of cooperative learning.
Tailor the work to reflect individual needs, interests, and styles.
Assign class work/homework over a range of abilities, allowing the students to complete various
problems as they feel capable of completing.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.






Begin with the hook.
Introduce essential questions.
Discuss uses of lesson and mathematical properties that apply.
Direct Instruction.
Homework – Practice skills and short writing task.
Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Mathematical Systems
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
Students will understand the qualities a set and an operation have to exhibit before becoming a
group. Then students will analyze other sets and operations to determine which are groups.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness;
Health Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 New Jersey Core Curriculum Standards including Cumulative Progress Indicator (CPI):


N-RN.3.
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number
and an irrational number is irrational; and that the product of a nonzero rational number and an
irrational number is irrational.
Unit Essential Questions:
What requirements do a set and an operation have to meet in order to meet the definition of a
group?
Does a certain set and a specific operation meet these requirements?
How is modular arithmetic used?
Unit Enduring Understandings:
The process of categorizing anything is a useful skill anywhere. Group theory is the basis for any
scientific method of classification.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 Discuss what requirements a set and an operation have to meet in order to meet the definition
of a group.


Decide if a certain set and a specific operation meet these requirements.
See how modular arithmetic is used.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessments
 Projects
Formative Assessments:
 Demonstration
 Homework
 Class Discussion
 Exit Ticket
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher: Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
Group theory is used by physicists and chemist us to study the behavior of the tiniest particles that
make up an atom. They use it in a field known as quantum mechanics. Group theory is used in the
science of robotics, computer graphics, weather forecasting, musical theory, and medical imaging.
Hook the learner with engaging work.
Have students use a Rubik's cube to discuss symmetry and transformations that could be used in
group theory.
Have students explore the relationship between permutation puzzles and group theory using the
book "Adventures In Group Theory" by David Joyner.
Have students study the science of cryptography by creating their own secret code, transmitting a
message, and then trying to break other codes.
Equip for understanding, experience and explore the big ideas.
Students will invent their own group using their own symbols in their set, and their own defined
operation.
Rethink opinions, revise ideas and work.
Students will work in groups to analyze other students' groups. Based upon observations they will
discuss items of concern.
Evaluate your work and adjust as needed.
Students will discuss their knowledge of group theory through cooperative learning activities. Any
questions and uncertainties will be discussed afterwards.
Tailor the work to reflect individual needs, interests, and styles.
Assign class work/homework over a range of abilities, allowing the students to complete various
problems as they feel capable of completing.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.
 Begin with the hook.
 Introduce essential questions.
 Discuss uses of lesson and mathematical properties that apply.
 Direct Instruction.
 Homework – Practice skills and short writing task.
 Self assessment questions.
HAMMONTON PUBLIC SCHOOLS
CURRICULUM PROJECT
Creating a Student-Centered Classroom
Content Area:
Mathematical Analysis, Applications, and Connections
Unit Title:
Systems of Numeration
Target Course/Grade Level:
12
School:
Hammonton High School
UNIT SUMMARY
The number system most of the world uses today, called the Hindu-Arabic system, is only one way to
communicate numerically. In this unit, other systems of numeration will be studied along with how to
perform basic arithmetic in other bases and with methods other than those most of us were taught
when we were children.
21st Century Skills:
Critical thinking and problem solving; Communication; Collaboration; Creativity and Innovation
21st Century Themes:
Civic Literacy; Financial, Economic, Business and Entrepreneurial Literacy; Global Awareness; Health
Literacy; Environmental Literacy
STAGE ONE: LEARNING TARGETS
2010 Common Core Curriculum Standards including Cumulative Progress Indicator (CPI):

N-RN.3. Explain why the sum or product of two rational numbers is rational; that the sum of a rational
number and an irrational number is irrational; and that the product of a nonzero rational number and an
irrational number is irrational.

Unit Essential Questions:
What is the history of our numerical system?
What evidence of other numerical systems do we have?
What other numerical systems exist? What are their symbols for quantities?
Why is it useful to convert between numerical systems?
Are there other ways to perform computations?
Unit Enduring Understandings:
Know the various systems of numeration. (Additive, multiplicative, ciphered, place-value)
Know the numerals used in Egyptian, Roman, Chinese, Greek, Babylonian and Mayan systems.
Understand the steps to convert between bases
Computational methods such as duplation, mediation, the lattice method, and Napier’s rods.
Key Knowledge and Skills students will acquire as a result of this unit:
Students will be able to …
 convert between different systems of numeration.
 convert between different bases.
 perform addition, subtraction, multiplication, and division in base 2.
 multiply using duplation and mediation.
 multiply using the Lattice method and Napier’s Rods.
STAGE TWO: EVIDENCE OF LEARNING
Summative Assessment:
 Quizzes
 Tests
 Quarterly Assessment
 Project
Formative Assessments:
 Homework Assignments
 Exit Ticket
 Demonstration
 Class Discussion
Student Self-Assessment and Reflection:
 Collins Writing Tasks
 Notebook
 Reflective Questioning
STAGE THREE: THE LEARNING PLAN
Sequence of teaching and learning experiences
Unit Resources:
 A Survey of Mathematics with Applications: Angel, Abbot, Rude; Publisher : Pearson
 LCD TV / Overhead Projector / Ipad
 Rulers / meter sticks
 White Board / Graph Board
 Graphing Calculators
 Geometer’s Sketchpad
Instructional Guidelines: Aligning Learning Activities, W.H.E.R.E.T.O.
Where is the unit headed?
This unit is designed to open the students’ horizons to other methods of representing quantities. Our
Hindu-Arabic system, while the most widely used numeration system, is not the only one in existence.
Exploring the history of numeration and other methods of computation will give students an
appreciation for what they have learned throughout grade school.
Hook the learner with engaging work.
Ask students to list how many different languages they know of. Then ask them to write down how
many different number systems they know of. Extend the discussion by showing examples of
Egyptian, Mayan, Greek, Chinese, Babylonian, and Roman Numerals
ASCII code used by computers to represent characters on the standard keyboard uses the last seven
positions of an eight-bit byte. How many different orderings of 0’s and 1’s can be made by using the
last seven positions of an eight-bit byte?
Equip for understanding, experience and explore the big ideas.
After exploring various ways to convert among systems of numerations and different bases, students
can be placed in groups to work on the U.S Postal Service Bar Codes project on pg 207 of the text.
Rethink opinions, revise ideas and work.
Revisit past assignments where conversions among number systems were taking place. Choose a
new number system and redo the conversion. Perform the same task where numbers were converted
between bases. Can you convert from base 2 to base 6 without converting to base 10 first?
Evaluate your work and adjust as needed.
What did you learn? Was there any information that was common sense to you or already known
before beginning this unit? Are any concepts still unclear? Can you create a study aid to organize all
of the different systems of numeration?
Tailor the work to reflect individual needs, interests, and styles.
Are there any other ancient methods to multiply or divide that you can discover through internet
research? Allow students a lab day to investigate other methods and have them prepare a lesson
worksheet, Power Point, or video that will explain how to perform the operation.
Organize the work flow to maximize in-depth understanding and success at the summative
tasks.
 Begin with the hook.
 Introduce essential questions.
 Discuss uses of lesson and mathematical properties that apply.
 Direct instruction.
 Homework – Practice skills and short writing task.
 Self assessment questions.